Subgroups — #28353375

Order 13851.

242 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#17987335	C₃	1	3	yes	maximal
2	C₃²×C₁₉×3₋¹⁺²	C₃	1	1	yes	maximal, characteristic
3	C₃²⋊#4294	—	3	3	no	—
4	C₃⁴×C₁₉	C₃²	1	1	yes	characteristic
5	C₃×C₁₉×3₋¹⁺²	—	3	1	no	—
6	C₃×C₁₉×3₋¹⁺²	—	3	1	no	—
7	C₃×C₁₉×3₋¹⁺²	—	3	9	no	—
8	C₃×C₁₉×3₋¹⁺²	C₉	1	3	yes	complemented
9	C₃²×C₉×C₁₉	—	3	1	no	—
10	#10624509	—	19	1	no	maximal, Sylow
11	C₃³×C₁₉	—	3	3	no	—
12	C₃³×C₁₉	—	9	3	no	—
13	C₃³×C₁₉	C₃×C₉	1	1	yes	characteristic
14	C₃³×C₁₉	H(3,𝔽₃)	1	1	yes	characteristic
15	C₃³×C₁₉	3₋¹⁺²	1	1	yes	characteristic
16	C₃³×C₁₉	3₋¹⁺²	1	1	yes	characteristic
17	C₁₉×3₋¹⁺²	—	3	9	no	—
18	C₁₉×3₋¹⁺²	—	3	9	no	—
19	C₁₉×3₋¹⁺²	—	3	9	no	—
20	C₃×C₉×C₁₉	—	3	9	no	—
21	C₃×C₉×C₁₉	—	3	3	no	—
22	C₃×#4294	—	9	3	no	—
23	C₃²×3₋¹⁺²	C₁₉⋊C₃	1	1	yes	characteristic
24	#6866	—	19	3	no	—
25	C₃²×C₁₉	—	3	3	no	—
26	C₃²×C₁₉	—	3	1	no	—
27	C₃²×C₁₉	—	3	1	no	—
28	C₃²×C₁₉	—	3	1	no	—
29	C₃²×C₁₉	—	9	1	no	—
30	C₃²×C₁₉	—	9	9	no	—
31	C₃²×C₁₉	—	9	1	no	—
32	C₃²×C₁₉	—	9	1	no	—
33	C₃²×C₁₉	C₃²⋊C₉	1	1	yes	characteristic
34	C₃²×C₁₉	AΣL(1,𝔽₂₇)	1	3	yes	—
35	C₉×C₁₉	—	3	9	no	—
36	#4294	—	27	3	no	—
37	C₃⁴	C₃×C₁₉⋊C₃	1	1	yes	characteristic
38	C₃×3₋¹⁺²	—	3	1	no	—
39	C₃×3₋¹⁺²	—	3	1	no	—
40	C₃×3₋¹⁺²	—	3	9	no	—
41	C₃×3₋¹⁺²	#4294	1	3	yes	complemented
42	C₃²×C₉	—	3	1	no	—
43	C₃²⋊C₉	—	57	3	no	—
44	C₃×C₁₉	—	3	3	no	—
45	C₃×C₁₉	—	3	3	no	—
46	C₃×C₁₉	—	3	1	no	—
47	C₃×C₁₉	—	9	1	no	—
48	C₃×C₁₉	—	9	1	no	—
49	C₃×C₁₉	#6866	1	1	yes	characteristic
50	C₃³	—	3	3	no	—
51	C₃³	—	9	3	no	—
52	C₃³	C₃×#4294	1	1	yes	characteristic
53	C₃³	#10587025	1	1	yes	characteristic
54	C₃³	#10587025	1	1	yes	characteristic
55	C₃³	C₃²⋊(C₁₉⋊C₃)	1	1	yes	characteristic
56	3₋¹⁺²	—	3	9	no	—
57	3₋¹⁺²	—	3	9	no	—
58	3₋¹⁺²	—	3	9	no	—
59	C₃×C₉	—	3	9	no	—
60	C₃×C₉	—	3	3	no	—
61	C₃×C₉	—	171	3	no	—
62	C₁₉	#10624509	1	1	yes	characteristic, complemented, Sylow
63	C₃²	—	3	3	no	—
64	C₃²	—	3	1	no	—
65	C₃²	—	3	1	no	—
66	C₃²	—	3	1	no	—
67	C₃²	—	9	1	no	—
68	C₃²	—	9	9	no	—
69	C₃²	—	9	1	no	—
70	C₃²	—	9	1	no	—
71	C₃²	C₃²⋊#4294	1	1	yes	characteristic
72	C₃²	#13095314	1	3	yes	—
73	C₉	—	513	3	no	—
74	C₉	—	3	9	no	—
75	C₃	—	3	3	no	—
76	C₃	—	3	1	no	—
77	C₃	—	3	3	no	—
78	C₃	—	9	1	no	—
79	C₃	—	9	1	no	—
80	C₃	#17987335	1	1	yes	characteristic

	subgroup	G/H
center	C₃	#17987335
commutator	C₃³×C₁₉	C₃×C₉
Frattini	C₃⁴	C₃×C₁₉⋊C₃
Fitting	C₃²×C₁₉×3₋¹⁺²	C₃
socle	C₃×C₁₉	#6866
prime radical, cosocle	C₃⁴×C₁₉	C₃²
hypercenter	C₃²×3₋¹⁺²	C₁₉⋊C₃
hypocenter	C₁₉	#10624509
LCS	G ▷ C₃³×C₁₉ ▷ C₃²×C₁₉ ▷ C₃×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₃ ◁ C₃³ ◁ C₃⁴ ◁ C₃²×3₋¹⁺²

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
8. C₃×C₁₉×3₋¹⁺²	C₉	semidirect
41. C₃×3₋¹⁺²	#4294	semidirect
62. C₁₉	#10624509	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#17987335	3
C₃²×C₁₉×3₋¹⁺²	3
#10624509	19